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February  2016, 36(2): 1023-1039. doi: 10.3934/dcds.2016.36.1023

## $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

 1 Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences, Wuhan 430071 2 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China 3 School of Mathematics, Wuhan University, 430072, Wuhan

Received  June 2014 Revised  March 2015 Published  August 2015

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.
Citation: Guji Tian, Qi Wang, Chao-Jiang Xu. $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1023-1039. doi: 10.3934/dcds.2016.36.1023
##### References:
 [1] L. Caffarelli, L. Nirenberg and J. Spruck, Dirichlet problem for nonlinear second order elliptic equations I,Monge-Ampère equations, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306. [2] B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math., 57 (2004), 1311-1331. doi: 10.1002/cpa.20010. [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0. [4] Q. Han, Local solutions to a class of Monge-Ampère equations of mixed type, Duke Math. J., 136 (2007), 421-474. [5] J. Hong and C. Zuily, Exitence of $C^{\infty}$ local solutions for the Monge-Ampère equation, Invent.Math., 89 (1987), 645-661. doi: 10.1007/BF01388988. [6] N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge-Ampère type, English transl., Math. USSR Sbornik, 50 (1985), 259-268. [7] N. M. Ivochkina, S. I. Prokofeva and G. V. Yakunina, The Gårding cones in the modern theory of fully nonlinear second order differential equations, Journal of Mathematical Sciences, 184 (2012), 295-315. doi: 10.1007/s10958-012-0869-1. [8] C. S. Lin, The local isometric embedding in $\mathbbR^3$ of 2-dimensional Riemannian manifolds with non negative curvature, Journal of Diff. Equations, 21 (1985), 213-230. [9] C. S. Lin, The local isometric embedding in $\mathbbR^{3}$ of two dimensinal Riemannian manifolds with Gaussian curvature changing sign clearly, Comm. Pure Appl. Math., 39 (1986), 867-887. doi: 10.1002/cpa.3160390607. [10] Q. Wang and C.-J. Xu, $C^{1,1}$ solution of the Dirichlet problem for degenerate k-Hessian equations, Nonlinear Analysis, T. M. A., 104 (2014), 133-146. doi: 10.1016/j.na.2014.03.016. [11] X.-N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbbR^3$, Journal of Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008. [12] X.-J. Wang, The $k$-Hessian equation, Geometric Analysis and PDEs, Lecture Notes in Mathematics, 1977 (2009), 177-252. doi: 10.1007/978-3-642-01674-5_5.

show all references

##### References:
 [1] L. Caffarelli, L. Nirenberg and J. Spruck, Dirichlet problem for nonlinear second order elliptic equations I,Monge-Ampère equations, Comm. Pure Appl. Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306. [2] B. Guan and J. Spruck, Locally convex hypersurfaces of constant curvature with boundary, Comm. Pure Appl. Math., 57 (2004), 1311-1331. doi: 10.1002/cpa.20010. [3] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0. [4] Q. Han, Local solutions to a class of Monge-Ampère equations of mixed type, Duke Math. J., 136 (2007), 421-474. [5] J. Hong and C. Zuily, Exitence of $C^{\infty}$ local solutions for the Monge-Ampère equation, Invent.Math., 89 (1987), 645-661. doi: 10.1007/BF01388988. [6] N. M. Ivochkina, A description of the stability cones generated by differential operators of Monge-Ampère type, English transl., Math. USSR Sbornik, 50 (1985), 259-268. [7] N. M. Ivochkina, S. I. Prokofeva and G. V. Yakunina, The Gårding cones in the modern theory of fully nonlinear second order differential equations, Journal of Mathematical Sciences, 184 (2012), 295-315. doi: 10.1007/s10958-012-0869-1. [8] C. S. Lin, The local isometric embedding in $\mathbbR^3$ of 2-dimensional Riemannian manifolds with non negative curvature, Journal of Diff. Equations, 21 (1985), 213-230. [9] C. S. Lin, The local isometric embedding in $\mathbbR^{3}$ of two dimensinal Riemannian manifolds with Gaussian curvature changing sign clearly, Comm. Pure Appl. Math., 39 (1986), 867-887. doi: 10.1002/cpa.3160390607. [10] Q. Wang and C.-J. Xu, $C^{1,1}$ solution of the Dirichlet problem for degenerate k-Hessian equations, Nonlinear Analysis, T. M. A., 104 (2014), 133-146. doi: 10.1016/j.na.2014.03.016. [11] X.-N. Ma and L. Xu, The convexity of solution of a class Hessian equation in bounded convex domain in $\mathbbR^3$, Journal of Funct. Anal., 255 (2008), 1713-1723. doi: 10.1016/j.jfa.2008.06.008. [12] X.-J. Wang, The $k$-Hessian equation, Geometric Analysis and PDEs, Lecture Notes in Mathematics, 1977 (2009), 177-252. doi: 10.1007/978-3-642-01674-5_5.
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